Liquid drops attract or repel by the inverted Cheerios effect

Abstract

Solid particles floating at a liquid interface exhibit a long-ranged attraction mediated by surface tension. In the absence of bulk elasticity, this is the dominant lateral interaction of mechanical origin. Here, we show that an analogous long-range interaction occurs between adjacent droplets on solid substrates, which crucially relies on a combination of capillarity and bulk elasticity. We experimentally observe the interaction between droplets on soft gels and provide a theoretical framework that quantitatively predicts the interaction force between the droplets. Remarkably, we find that, although on thick substrates the interaction is purely attractive and leads to drop-drop coalescence, for relatively thin substrates a short-range repulsion occurs, which prevents the two drops from coming into direct contact. This versatile interaction is the liquid-on-solid analog of the "Cheerios effect." The effect will strongly influence the condensation and coarsening of drops on soft polymer films, and has potential implications for colloidal assembly and mechanobiology.

Keywords: droplets; elastocapillarity; mechanosensing; soft matter; wetting.

Fig. 1.

The inverse Cheerios effect for droplets on soft solids. Two liquid drops sliding down a soft gel exhibit a mutual interaction mediated by the elastic deformation of the substrate. (A) Drops sliding down a thick elastic layer attract each other, providing a new mechanism for coalescence. (B) Drops sliding down a thin elastic layer (thickness $h_0$) repel each other. (C) Measurement of the horizontal relative velocities $\mathrm{\Delta }{v}_x$ of droplet pairs, as a function of separation distance d. This measurement quantifies the interaction strength. (D) Sliding velocity of isolated droplets on the thick layer as a function of their volume. These data are used to calibrate the relation between force (gravity) and sliding velocity.

Fig. 2.

Measured interaction force F (symbols) as a function of their separation d, compared with the 3D theory (red lines, no adjustable parameters). (A) Attraction on a thick elastic layer ($h_0\approx 8\hspace{0.5em}\text{mm}\gg R\gg \ell$). (B) Repulsion and attraction on a thin layer ($R\gg \ell \gtrsim h_0\approx 40\hspace{0.5em}\mathrm{\mu }\text{m}$). Each data point represents an average over ∼10 realizations, with the error bars giving the SD. Measurements are based on pairs of ethylene glycol drops whose radii are in the range $R\sim 0.7\pm 0.1\hspace{0.5em}\text{mm}$. The elastic substrate has a static shear modulus of $0.28\hspace{0.5em}\text{kPa}$.

Fig. 3.

Mechanism of interaction between two liquid drops on a soft solid. (A) Deformation $h\left(x\right)$ induced by a single droplet on a thick substrate. The zoom near the contact line illustrates that the contact angles satisfy the Neumann condition. (B) A second drop placed on a thick substrate experiences a background profile due to the deformation induced by the neighboring drop on the right. This background profile is shown in red. As a consequence, the solid angles near the elastic meniscus rotate by an angle φ (see zoom). This rotation perturbs the Neumann balance, yielding an attractive force f. In the experiment, this force is balanced by the dissipative force due to the viscoelastic deformation of the wetting ridge. (C) On a thin substrate the single-drop profile yields a nonmonotonic elastic deformation. The zoom illustrates a rotation φ of the Neumann triangle in the opposite direction, leading to a repulsive interaction.

Fig. S1.

Problem setup. Two liquid droplets on an elastic half space. Energy is to be minimized with respect to $\mathrm{\mathscr{H}}\left(x\right)$, $h_{sv}\left(x\right)$, $h_{sl}\left(x\right)$, $x_i$, and $x_o$. The liquid angles ${\theta }_i$ and ${\theta }_o$ follow from the minimization. The distance d between the adjacent contact lines is enforced with a Lagrange multiplier f acting on $\left(x_i-d/2\right)$.

Fig. S2.

Force–distance curve for 2D droplets on elastic half space (log–log scale), for $\gamma /\left(GR\right)=1/\sqrt{10}$, $\gamma /{\gamma }_s=\sqrt{10}$. The Inset shows solid profiles at the indicated distances.

Fig. 4.

Three-dimensional calculation of interface deformation for a pair of axisymmetric drops. The elastocapillary meniscus between the two drops is clearly visible, giving a rotation of the contact angle around the drop. The total interaction force $\overrightarrow{F}$ is obtained by integration of the horizontal force $\overrightarrow{f}$ (indicated in red) and is related to the free-energy gradient associated with a change in separation between the drops. Parameter values are $\ell /R=0.1$, $\gamma /{\gamma }_s=1$.

Fig. S3.

Problem setup for axisymmetric drops on an elastic half space. The solid shape is shown for a distance $d=0.3R$ between the droplets and $\gamma /{\gamma }_s=10$. x and y are in units of R, and z is in units of $\gamma /G$.

Fig. S4.

Force–distance curve for axisymmetric droplets on a thin elastic layer. (Inset) $f_r\left(\beta \right)$ at the contact line of a droplet on a thin layer in the presence of a second droplet at $d/h_0=2$. $h_0/R=1/20$, $\gamma /\left(GR\right)=1/\sqrt{10}$, $\gamma /{\gamma }_s=\sqrt{10}$.

Fig. S5.

Spectra of storage ($G\prime$) and loss modulus ($G″$) of the substrates. Curves for −10 °C and 80 °C were shifted to obtain a room temperature master curve.